The study of statistical efficiency under LDP constraints has gained considerable traction, e.g., [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}. Almost all of these works consider \(\varepsilon \) -LDP and provide meaningful bounds
only for sufficiently small values of \(\varepsilon \)  (i.e., the high privacy regime). For instance, Duchi et al. [2]} studied minimax estimation problems under \(\varepsilon \) -LDP constraints and showed that for \(\varepsilon \le 1\) , the price of privacy is to reduce the effective sample size from \(n\)  to \(\varepsilon ^2 n\) . A slightly improved version of this result appeared in [13]}, [5]}. More recently, Duchi and Rogers [15]}
developed a framework based on the strong data processing inequality (SDPI) [16]} and derived lower bounds for minimax estimation risk under \(\varepsilon \) -LDP that hold for any \(\varepsilon \ge 0\) .
